What a passing wave does to a ring

At its defaults it draws what a passing wave does to a ring. A ring of 8 free masses at 4 phases of a passing gravitational wave, in both polarisations. The upper row is the + mode: one diameter lengthens while the perpendicular one shortens, and half a cycle later they swap. The lower row is the × mode, which is the same pattern rotated by forty-five degrees rather than ninety — the signature of a spin-2 field, and the reason a detector is built as two arms at a right angle. The drawn strain is 0.42; a real one is 10⁻²¹, so the deformation is exaggerated 4.2·10²⁰ times. At that true strain a four-kilometre arm changes length by 4·10⁻¹⁸ m.

quadrupole-wave is one function in lib/figures/gravity.js — geometry, horizons, and the waves in it. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page. A figure here is the figure a reader meets in an essay, and if the generator changes, this page changes with it.

A ring of 8 free masses at 4 phases of a passing gravitational wave, in both polarisations. The upper row is the + mode: one diameter lengthens while the perpendicular one shortens, and half a cycle later they swap. The lower row is the × mode, which is the same pattern rotated by forty-five degrees rather than ninety — the signature of a spin-2 field, and the reason a detector is built as two arms at a right angle. The drawn strain is 0.42; a real one is 10⁻²¹, so the deformation is exaggerated 4.2·10²⁰ times. At that true strain a four-kilometre arm changes length by 4·10⁻¹⁸ m.

The strain radiated by a remnant of 62 solar masses spinning at 0.68 of its maximum, as it settles down, with the decaying envelope of its fundamental mode drawn over it. The fundamental rings at 274 hertz and decays in 3.7 milliseconds, which is 1.0 cycles — this is not a bell and it does not sustain. Every frequency and every decay time in the sum is fixed by the mass and the spin alone; nothing about what made the remnant survives into them. What does depend on the collision is how loudly each mode is excited, and the relative amplitudes here are the rough values a merger of two comparable masses produces rather than a prediction.

The quasinormal frequencies as points in the complex plane — the pitch across, the decay rate up — in units of the inverse mass, with the track each mode follows as the spin runs from zero to nearly its maximum. Every track runs right and down: a faster-spinning hole rings higher and rings for longer. The approach to the bottom right is the extremal limit, where the decay rate goes to zero and the ring would never stop, and it is approached rather than reached — as the spin cannot exceed one, so this corner of the picture is the boundary of what a black hole is allowed to be. The points at zero spin are stacked in pitch by their angular index and are all at nearly the same height, because at zero spin the decay rate barely depends on which mode it belongs to.

The frequencies of 3 quasinormal modes against the remnant's spin, each divided by the fundamental's frequency at zero spin, for a mass of 62 solar masses. Changing the mass slides all of them together, in inverse proportion, and changes nothing about their ratios. So the whole spectrum is one curve per mode in one variable, the spin — which is what the no-hair theorem amounts to as a statement about a measurement, and why a second mode is worth so much more than a louder first one. A hole at 90 per cent of maximum spin rings 1.81 times higher in its fundamental than a non-spinning one of the same mass, so the pitch alone cannot separate mass from spin, and two quantities read off one mode — its pitch and how fast it dies — can.

What a ringdown measurement says about the remnant, drawn in the plane of its mass and its spin. A single number — one mode's pitch — fixes not a point but a curve, because two unknowns need two measurements. How fast that same mode dies gives a second curve, and the two cross: one mode measured in both its parts already determines the mass and the spin. The other two curves are the same pair for the ℓ=3, m=3, fundamental. The four cross at one point only because the remnant really is a Kerr black hole. That is the whole of black-hole spectroscopy: the first mode measures, and every mode after it tests, because nothing in the theory allows a second frequency to be anything but a function of the mass and spin the first one gave. A remnant with any other property — a surface, an exotic interior, extra structure of any kind — would put the curves through different places.

What a ringdown measurement says about the remnant, drawn in the plane of its mass and its spin. A single number — one mode's pitch — fixes not a point but a curve, because two unknowns need two measurements. How fast that same mode dies gives a second curve, and the two cross: one mode measured in both its parts already determines the mass and the spin. The other two curves are the same pair for the ℓ=2, m=2, first overtone. The four cross at one point only because the remnant really is a Kerr black hole. That is the whole of black-hole spectroscopy: the first mode measures, and every mode after it tests, because nothing in the theory allows a second frequency to be anything but a function of the mass and spin the first one gave. A remnant with any other property — a surface, an exotic interior, extra structure of any kind — would put the curves through different places.

physicscheck asserts something about quadrupole-wave that could fail — it draws it and measures the result against a value reached some other way.

Across the library: 100 interrogated, 2 exercised only, 1 untouched, of 103. Read out of the gate's source by the gate's own two patterns — and the gate's last claim fails the build if that read disagrees with what it was handed while running.